On Hilbert’s irreducibility theorem
Volume 180 / 2017
Acta Arithmetica 180 (2017), 1-14
MSC: 11C08, 11G35, 11R32, 11R45.
DOI: 10.4064/aa8380-2-2017
Published online: 9 August 2017
Abstract
We obtain new quantitative forms of Hilbert’s irreducibility theorem. In particular, we show that if $f(X, T_1, \ldots, T_s)$ is an irreducible polynomial with integer coefficients, having Galois group $G$ over the function field $\mathbb Q(T_1, \ldots, T_s)$, and $K$ is any subgroup of $G$, then there are at most $O_{f, \varepsilon}(H^{s-1+|G/K|^{-1}+\varepsilon})$ specialisations $\mathbf{t} \in \mathbb Z^s$ with $|\mathbf{t}| \le H$ such that the resulting polynomial $f(X)$ has Galois group $K$ over the rationals.
